91Ó°ÊÓ

Gauge pressure is essential for understanding how hydraulic systems work. It is the difference between the absolute pressure and atmospheric pressure. In this exercise, we need to find the gauge pressure required to lift a car using a hydraulic automobile lift. The formula for calculating pressure is:\[ P = \frac{F}{A} \]where \( P \) is pressure, \( F \) is the force exerted by the car's weight, and \( A \) is the area of the piston's cross-section.

This means that once we calculate the force required to lift the car and the area of the piston, we can determine the necessary gauge pressure in pascals. Understanding this helps us manipulate properties of fluids effectively in hydraulic systems.

In hydraulic systems, the relationship between force and area is a cornerstone concept. Force affects how much pressure is exerted onto an area. The car’s weight, translated into force, is what we need to calculate first using the formula:\[ F = m \times g \]where \( m \) is the mass of the car, and \( g \) is the acceleration due to gravity.

In this example, \( m = 1200 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \), giving us a force of \(11772 \text{ N}\). Next, calculating the area of the circular piston using:\[ A = \pi \times r^2 \]where \( r = 0.15 \text{ m} \). This relationship allows us to understand how force distributed over a specific area affects pressure in hydraulic systems.

Pressure conversion is crucial for standardizing measurements across different units. In this context, gauge pressure is initially calculated in pascals. However, expressing it in atmospheres can provide a better perspective for practical applications. To convert pascals to atmospheres, use the conversion factor:

• 1 atmosphere (atm) = 101325 pascals (Pa)

The pressure in our solved problem was found to be \(166564.8 \text{ Pa}\). Converting this to atmospheres involves:\[ P = \frac{166564.8}{101325} \approx 1.644 \text{ atm} \]

Converting units helps in comparing pressures easily under different contexts, making calculations more versatile.

Understanding the relationship between weight and gravity is essential in mechanics. Weight is essentially a force and is calculated by multiplying mass with the acceleration due to gravity. This is central to many physical concepts, especially in hydraulic systems like the lift in our example.

Here, the car's mass is given as \(1200 \text{ kg}\), and the standard gravity is typically \(9.81 \text{ m/s}^2\). Thus, the weight of the car can be found using:\[ F = m \times g = 1200 \times 9.81 = 11772 \text{ N} \]

This relationship illustrates how gravitational force interacted with mass creates the weight necessary to apply pressure in hydraulic systems. It shows the critical step of converting mass, a scalar quantity, into force, a vector quantity, thereby allowing us to proceed with hydraulic pressure calculations effectively.